@Fidge Although purists would agree with you, current usage of 'got' in this construction is very common. And since it's the educated speakers who ultimately decide the correctness of anything, using 'got' isn't wrong. :)
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mikhailcaziFeb 20 '14 at 10:14

6 Answers
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I would say it's not incorrect grammatically speaking, but perhaps imprecise. It would be clearly incorrect to say something like "Have you got pencil?"

'Paper', on the other hand, in the sense you've used it is a mass noun. The usage in English seems a bit murky, but you'd be well understood in a classroom if you asked a child "Have you got paper?" in the same way as if you asked "Have you got money?"

I've deleted my own answer because you've added the relevant fact that in this context, paper is a mass noun. But suppose the teacher had preceded the question with "For this lesson I want you each to paint two pictures - one seascape, and one landscape". In that case she'd expect any pupils who have only one sheet of paper to speak up. Effectively, the question means "Have you gotenoughpaper?"
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FumbleFingersDec 1 '11 at 23:29

@FumbleFingers, you're spot on, it's a question of context, and there are certainly lots of circumstances where one could add an extra word in there for clarity.
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SnubianDec 1 '11 at 23:41

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I've seen several questions like this lately, where the only reason some particular variation occurs less often is because the contexts in which it would be suitable are less common. But people post answers saying the uncommon version is "wrong", and they get upvoted by others who simply don't stop to consider all possible contexts. It's totally different to those questions where OP defines exactly what context he has in mind - although even there we must be careful about saying some less common variant is actually wrong.
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FumbleFingersDec 1 '11 at 23:59

Yes, it is grammatical. It shows a use of the zero article, of which the ‘Longman Student Grammar of Spoken and Written English’ says, Zero article phrases commonly express non-specific or generic references. To see that ‘paper’ can occur with a zero article, we need look no further than Lewis Carroll:

I see that "have you got paper?" is not the same as ...a piece of paper, ...any paper, or ...some paper.

In the first case, I am concerned about paper, (the right kind in the right quantity, which is supposed to be understood) that would serve my purpose. All other forms of the sentence distract (from) the focus. I will be inviting trouble in the form of "I have lots of paper, but sorry no pieces"; "Would any paper really do?"; "How much, according to you, is some paper?"

So long as grammar lets me, I will focus on the core idea and leave the peripheral details unsaid. :)